ICES REPORT 14-20

August 2014

Fluid-Filled Fracture Propagation using a Phase-FieldApproach and Coupling to a Reservoir Simulator

by

Thomas Wick, Gurpreet Singh, Mary F. Wheeler

The Institute for Computational Engineering and SciencesThe University of Texas at AustinAustin, Texas 78712

Reference: Thomas Wick, Gurpreet Singh, Mary F. Wheeler , "Fluid-Filled Fracture Propagation using aPhase-Field Approach and Coupling to a Reservoir Simulator," ICES REPORT 14-20, The Institute forComputational Engineering and Sciences, The University of Texas at Austin, August 2014.

Fluid-Filled Fracture Propagation using a Phase-Field

Approach and Coupling to a Reservoir Simulator

Thomas Wick Gurpreet Singh Mary F. Wheeler

July 31, 2014

Abstract

Tight gas and shale oil play an important role in energy security and in meeting anincreasing energy demand. Hydraulic fracturing is a widely used technology for recov-ering these resources. A quantitative assessment of hydraulic fracturing jobs relies uponaccurate predictions of fracture growth during slick water injection for single and mul-tistage fracturing scenarios. It is also important to consistently model the underlyingphysical processes from hydraulic fracturing to long-term production. A recently intro-duced thermodynamically consistent phase-field approach for pressurized fractures inporous medium is utilized which captures several characteristic features of crack propa-gation such as joining, branching and non-planar propagation in heterogeneous porousmedia. This phase-field approach captures both the fracture-width evolution and thefracture-length propagation; here we address the latter. In this work, we briefly describeour phase-field fracture propagation model and then present a technique for couplingthis to a fractured poroelastic reservoir simulator. The proposed coupling approach canbe adapted to existing reservoir simulators. We presents two and three dimensionalnumerical tests to benchmark, compare and demonstrate the predictive capabilities ofthe fracture propagation model as well as the proposed coupling scheme.

Keywords. phase-field; pressurized fractures; iterative coupling algorithm; reservoirsimulation

Introduction

Hydraulic fracturing is a well known method for recovering oil and gas from tight gas andshale plays. It is pivotal in meeting a continually growing energy demand. Concerns are alsobeing raised regarding its impact on long and short term environmental implications. Thus,there is an imminent need for physically and mathematically consistent, accurate and robustcomputational models for representing fluid field fractures in a poroelastic medium. Thesimplest model description involves coupling of (1) mechanical deformation, (2) reservoir-fracture fluid flow, (3) and fracture propagation. The rock deformation is usually modeledusing the linear elasticity theory (Biot (1941a,b, 1955)). For fluid flow modeling, lubrica-tion theory and Darcy flow are assumed in the fracture and reservoir respectively, which arecoupled through a leakage term. Finally, for fracture propagation the conventional energy-release rate approach of linear elastic fracture mechanics (LEFM) theory is used. We alsonote some of the concurring modeling and numerical approaches for fracture propagationcurrently used such as cohesive zone models (Xu and Needleman (1994)), displacement dis-continuity methods (Crouch (1976)), partition-of-unity (Babuska and Belenk (1997)) based

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XFEM/GFEM methods (Moes et al. (1999); de Borst et al. (2006); Secchi and Schrefler(2012); Babuska and Banerjee (2012)), boundary element formulations (BEM) (Castonguayet al. (2013)) and peridynamics (Silling (2000)).

Variational approaches (Francfort and Marigo (1998); Bourdin et al. (2008)) and athermodynamically consistent phase field formulation (C. Miehe (2010)) have been employedin solid mechanics. An application to hydraulic fracturing is given in (Bourdin et al. (2012)).C. Miehe (2010) extended the variational approach (Francfort and Marigo (1998); Bourdinet al. (2008)) by modeling crack irreversibility through an entropy condition satisfying thesecond law of thermodynamics and decomposing the strain tensor to account for tension andcompression. Our approach (Mikelic et al. (2013a,b)) is based upon C. Miehe (2010) withan extension to porous media applications where solids (geomechanics) interact with fluids.To develop a phase-field formulation for such applications, geomechanics and porous mediaflows are decoupled using fixed-stress splitting (Settari and Walters (2001); Mikelic andWheeler (2012)). With this methodology, modeling and simulations of hydraulic fracturesin poroelasticity have been considered (Mikelic et al. (2013a,b, 2014b); Wheeler et al. (2014);Mikelic et al. (2014a); Wick et al. (2014)).

We provide a brief recapitulation describing Griffith’s model for fracture growth inbrittle media. The classical theorem of minimum energy suggests that an equilibrium stateachieved by an elastic body deformed by surface forces is such that the potential energyof the system is minimal. This was later augmented by Griffith (1921), in his seminalwork, assuming a different equilibrium state is possible which accounts for formation offractures as a mechanism for lowering potential energy. This criterion of rupture assumesthat the cohesive forces, due to molecular attraction, act close to the fracture tip. Thus,the contribution of cohesive forces (surface potential energy) to the total potential energycan be assumed to be negligible. Based upon these assumptions, a decrease in potentialenergy is proportional to the generated surface area with the critical energy release rate(Gc) as the constant of proportionality. In this work, we rely upon this classical workalong with its assumptions on the fracture growth criteria. As noted by Barenblatt (1962),we do not underestimate the significance of cohesive forces at the fracture tip. However,the contribution of these forces to the total potential energy is assumed to be significantonly during fracture nucleation which diminishes as the fracture grows. Based upon thesearguments we assume that LFEM is applicable.

Using a phase-field approach, a lower-dimensional crack surface is approximated as adiffusive transition zone by a phase-field function ϕ. Fig. 1 shows this diffusive transitionzone (also brittle or mushy-zone) between the broken (white zone) and the unbroken (brownzone) states of the material. A fixed-topology finite element phase-field approach is shownwhere a (lower-dimensional) crack is approximated with the help of a phase-field function.The phase-field function is an indicator function with values 0 and 1 inside and outside thecrack, respectively. The mushy-zone also provides a smooth interpolation for the interfacebetween a fracture and reservoir. A coupling of reservoir fluids and geomechanics allows acomprehensive study of this multiscale problem where only few results have been publishedto date (see for instance Dean and Schmidt (2008) and Lujun and Settari (2007)). Further,we also describe an algorithm to integrate fracture growth patterns with our reservoir sim-ulator IPARS (Implicit Parallel Accurate Reservoir Simulator). This allows for both shortterm transient pressure analysis and long term recovery predictions. We note that crack or

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fracture propagation, which will be used interchangeably, implies both variation of fracturewidth (or aperture) and its length.

The major advantages of using phase-field modeling for crack propagation are four-fold: First, and most important, the model is easy to implement and uses a fixed-gridtopology in which remeshing for resolving the exact fracture location is avoided. Second,fracture nucleation, propagation, kinking, and curvilinear path are intrinsically determined.This avoids computational overheads associated with post-processing of quantities such asstress intensity factors. Third, we can easily handle large fracture networks since complexphenomena of joining and branching does not require keeping track of fracture interfaces.Fourth, modeling crack growth in heterogeneous media does not require special treatment.Here however, the length-scale parameter ε should be chosen accordingly. Additionally,the crack opening displacement (fracture aperture) can be calculated using the phase-fieldfunction. We use the pressurized crack propagation model in a poroelastic medium using aphase-field approach proposed by Mikelic et al. (2013a,b).

Figure 1: Evolution of two pressurized fractures: first joining, then nonplanar growth andfinally branching in heterogeneous porous media.

Our focus in this paper is on the following aspects. We investigate the previouslydescribed phase-field approach for different crack propagation scenarios including hetero-geneous porous media including permeability and geomechanical parameters. Second, weperform detailed studies of multi-stage fractures. Third, we consider the phase-field modelas a fractured-well approach in a reservoir and we consequently couple this approach to areservoir simulator. This paper concentrates primarily on the approach for fracture growthusing slick-water injection. We account for varying reservoir complexities such as natu-ral fractures, faults and barriers using a comprehensive fractured poroelastic reservoir flowmodel. This allows for a two-stage production optimization owing to (1) a well-engineeredhydraulic fracturing scheme followed by (2) an optimal fractured well placement consideringfar from well-bore reservoir complexities. The paper is organized as follows: we first pro-vide our motivation for the work and the reason for our choice of using a phase-field modelfor hydraulic fracturing. In the next section, we provide the governing equations for thefracture phase-field approach the reservoir flow equations. In the section after, we providedetails on the coupling algorithm between the fracture phase-field model and the reservoirsimulator. In the final section, numerical tests are discussed to demonstrate our method.

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Pressurized and Fluid-filled Crack Propagation Models using Phase-field

In this section, we describe the proposed model development starting by defining a two-field problem in two unknowns: (1) a vector displacement field and (2) a scalar phase-fieldvariable (ϕ), assuming a known pressure field (the so-called pressurized fracture propaga-tion). This is later extended to a three-field problem, adding scalar pressure (p) as anunknown, accounting for flow inside the porous rock matrix and the fracture (the so-calledfluid-filled fracture propagation). Therein, a single pressure diffraction equation (see Mikelicet al. (2014a)), derived from the mass conservation equation and Darcy’s law, is used forlocal flow field calculations. The elasticity and phase-field equations are formulated as anenergy minimization problem. We obtain a weak form of the differential equations by dif-ferentiating this energy minimization function with respect to the solution variables. Thisserves as a natural setting for using a Galerkin finite element method for spatial discretiza-tion. Before we begin, it is important to discuss the key features of the classical brittlefracture theory (Griffith (1921)) used in this work. The theory postulates two physicalphenomena: (1) linear elasticity and (2) fracture propagation, as energy dissipation mech-anisms, strictly separated by a threshold (critical energy release rate) assuming a sharptransition between the fractured and non-fractured media. The Griffith’s criterion for brit-tle fracture propagation assumes:

1. The crack growth is irreversible.

2. The energy release rate is bounded above by a critical energy release rate.

3. The crack grows if and only if the energy release rate is critical.

Let Ω be the reservoir domain, as shown in Fig. 2, with the fracture C ⊂ Ω.

Ω

C

Figure 2: Problem description.

Pressurized fracture propagation model (two-field problem) We begin by describ-ing the pressurized fracture approach where a known pressure field is assumed on the domainΩ. The pressure remains invariant along the fracture length varying only temporally basedupon a simple correlation. The pressure variation along the fracture length is assumed tobe negligible. Later we show that this assumption is valid if the fracture conductivity is

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substantially larger than the reservoir conductivity, which is usually the case. The energyfunctional for a poroelastic material (Ω) with a crack (C) reads:

E(u, C) =1

2

∫ΩσE : e(u)︸ ︷︷ ︸

Elastic energy

−∫

ΩαB(p− p0)∇ · u︸ ︷︷ ︸

Pore pressure contribution

+ GcHd−1(C)︸ ︷︷ ︸Fracture energy

, (1)

with the following constitutive stress-strain equation and definition of strain e(u):

σE = 2µe(u) + λtr(e(u))I, (2)

e(u) =1

2(∇u+∇uT ). (3)

where µ and λ denote the Lame parameters, σE the Cauchy stress tensor, e(u) thestrain tensor, αB the Biot coefficient, p the pore pressure, po the reference pressure, u thedisplacements. Gc is the critical elastic energy release rate depending on the material andis determined experimentally and Hd−1 is the length of the fracture. Please note that Gcis related to stress intensity factor under certain assumptions on the material such as anisotropic, linear elastic solid (Irwin (1958)). Further, we follow the approach presentedby Ambrosio and Tortorelli (1990) for approximating the fracture length (Hd−1) using anelliptic functional

1

2ε‖1− ϕ‖2 +

ε

2‖∇ϕ‖2, (4)

thereby introducing an additional variable ϕ, referred to as the phase-field variablehereafter. This variable is a quantity defined on the entire domain Ω for a time spanvarying from 0 to T. A careful examination of Eqn. 4 shows, for a given value of ε > 0, thisfunctional assumes lowest values when ϕ is a constant assuming values of either 0 (fracture)or 1 (porous rock matrix). We notice that the phase-field approach is related to gradient-type material modeling with a characteristic length-scale. Here, the above regularizationparameter ε can be considered as such a length-scale parameter that has a physical meaning(Pham et al. (2011); C. Miehe (2010) and references cited therein). The second term ensuresthat ϕ changes smoothly between 0 and 1 allowing the representation of the fracture as adiffuse interface. Eqn. 4 represents a mathematically-consistent approximation of the truecrack Hd−1.

In order to satisfy assumptions 2 and 3 the energy functional (Eqn. 1) is regularizedwith ϕ as follows:

Eε(u, ϕ) =1

2

∫Ω

((1− κ)ϕ2+ + κ)σE : e(u)−

∫ΩαB(p− p0)ϕ2

+∇ · u

+Gc

(1

2ε‖1− ϕ‖2 +

ε

2‖∇ϕ‖2

)(5)

Here, ϕ+ is the maximum of ϕ and 0, κ ≈ 0 (determined by machine precision) is a positiveregularization parameter for the elastic energy and the length-scale parameter ε denotes thewidth of the transition zone in which ϕ changes from 0 to 1 (this width is illustrated as the

5

contour lines between the white and brown regions in Fig. 1). One can see from Eqn. 5that if ϕ is 0 (fracture), the first and second terms become zero and the energy functionalis dominated by the critical energy release rate Gc. Similarly when ϕ is 1, the third termbecomes zero. An intermediate behavior can be seen for values between 0 and 1. Finally,we impose the irreversibility constraint (assumption (1)) on ϕ; i.e.,

∂tϕ ≤ 0, (6)

which ensures that the state variables change in the direction of energy minimization orentropy maximization, in accord with the 2nd law of thermodynamics. Then, the finalenergy functional reads:

Eε(u, ϕ) =1

2

∫Ω

((1− κ)ϕ2+ + κ)σE : e(u)− 1

2

∫ΩαB(p− p0)ϕ2

+∇ · u

+Gc

(1

2ε‖1− ϕ‖2 +

ε

2‖∇ϕ‖2

)+ IK(ϕn−1)(ϕ), (7)

where the last term IK(ϕn−1)(ϕn) is a penalization term to impose the irreversibility con-

straint (6). We are now ready to derive differential equations in a Galerkin fashion that canbe easily adapted and implemented in legacy reservoir simulators using finite element dis-cretizations. For the sake of brevity, we directly introduce these differential equations. Thereader is referred to Mikelic et al. (2013a) for a detailed derivation. The problem statementthen reads: Find u and ϕ such that,∫

Ω

((1− κ)ϕ2

+ + κ)Ge(η) : e(w)−

∫Ω

(αB − 1)(ϕ2+pdiv w) +

∫Ωϕ2

+∇pw = 0

∀ admissible test functions w,

as well as,∫Ω

(1− κ)(ϕ+ Ge(η) : e(η)ψ −∫

Ω2(αB − 1)(ϕ+ p div η)ψ + 2

∫Ωϕ+∇p ηψ

+Gc

(−∫

Ω

1

ε(1− ϕ)ψ +

∫Ωε∇ϕ∇ψ

)+

∫Ω

(Ξ + γ(ϕ− ϕn−1))+ψ = 0

∀ admissible test functions ψ.

Here, Ξ and γ are a penalization function and parameter, respectively, to enforce the irre-versibility constraint of crack growth with the help of an augmented Lagrangian formulation(Wheeler et al. (2014)). In the last term, ϕn−1 denotes the phase-field solution to the pre-vious time step.

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Fluid-filled fracture propagation model (three-field problem) In the previous sec-tion, a given uniform fracture pressure was assumed for crack propagation. Here we brieflydescribe an extension of this approach where a pressure field is computed by solving aflow problem on the entire domain (both reservoir and fracture). An extended Reynold’slubrication equation and Darcy’s law are solved in the fracture and reservoir domains, re-spectively along with the fluid mass conservation equations. The benefit of our proposedapproach is that both sets of equations have similar structure identified by the phase-fieldvariable as separate fracture (ΩF (t)) and reservoir (ΩR(t)) domains. Here, ΩF (t) is thevolume approximation of the crack C. For further details the reader is referred to Mikelicet al. (2014a). The mass conservation equations for fluid flow are:

∂tρF +∇ · (ρF vF ) = qF − qL in ΩF (t),

∂t(ρRφR) +∇ · (ρRvR) = qR in ΩR(t).(8)

Please note that the fracture porosity is set to one. The fracture volume is accounted forby the spatial discretization. Here, the velocities are defined by the Reynold’s lubricationequation and Darcy’s law for the fracture and the reservoir, respectively:

vj = −Kj

νj(∇pj − ρjg). (9)

The term qL represents the leakage from the fracture owing to the 3D approximation of2D Reynold’s lubrication equation for the fracture domain. Where, j = F,R denotes thefracture and reservoir domains, φj the fluid fraction, Kj the permeability tensor, νj andρj the fluid viscosity and density, respectively, g the gravity and qj the source/sink term.

A comparison of Darcy’s law and Reynold’s lubrication equation shows that KF = w(u)2

12µ ,where w(u) is the fracture width (or aperture) calculated from jump in normal displacementsu. A detailed derivation of our leakage term can be found in Mikelic et al. (2014a).

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Discretization and solution algorithm The flow and mechanics equations are solvedusing the fixed-stress iterative coupling scheme (Settari and Walters (2001); Mikelic andWheeler (2012)) where decoupling is achieved as illustrated in Figure 3 along with thealgorithmic flow chart 1. We then first discretize in time using a backward Euler schemefollowed by spatial discretization with a continuous Galerkin finite element method on ahexahedral grid with grid size parameter h. Here, all variables are discretized by continuousbilinears in space. We note that h ε, which requires fine meshes around the fracture(s).To this end, we use local mesh refinement with hanging nodes (see Figure 4).

Algorithm 1 Augmented Lagrangian fixed-stress solution algorithm

For each time tn

repeatSolve augmented Lagrangian loop (outer loop)repeat

Solve two-field fixed-stress (inner loop):Solve the pressure diffraction Problem (8)Solve linear elasticity in Problem (8)

until Stopping criterion

max‖ul − ul−1‖, ‖pl − pl−1‖ ≤ TOLFS, TOLFS > 0

for fixed-stress split is satisfiedSolve the nonlinear phase-field in Problem (8)

UpdateΞk+1 = (Ξk + γ(ϕk+1 − ϕn−1))+, k = 0, 1, 2, . . .

until Stopping criterion

‖Ξk−1 − Ξk‖ ≤ TOLAL, TOLAL > 0

is satisfiedSet: (un, ϕn) := (uk, ϕk).Increment tn → tn+1.

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pl

ul

ϕl

pl+1

ΩF

Given ul−1, ϕl−1

Pressure force in elasticity

Pressure force in phase-field

α and w

Bulk energy coefficient

ul+1

Previous step Next stepPresent step 0 Present step 1 Present step 2

Compute pl Compute ϕlCompute ul

Bulk energy term

Figure 3: Explication how the three equations for pressure, displacements and phase-fieldcouple together. In each iteration step (within each time step), we compute first pl, thenul, and finally ϕl, where each variable influences the other two. Specifically, ϕl alters thefracture domain ΩF, which influences the computation of the new pressure pl+1.

Integrating Phase-field Crack Propagation and Fractured Reservoir Flow Models

In this section, we describe the proposed coupling method while outlining a work-flowfor translating fracture location, geometry and width information between the phase-fieldcrack propagation model and the production reservoir code. The use of hexahedral elementsfor spatial discretization in both models allows translation of fracture location and variablesfrom one model to another. The phase-field with crack growth and localized flow is used asa pre-processor step for the fractured reservoir flow. This results in a forward solution withthe pertinent fracture geometry and width translated at the end of the propagation. Weconsider phase-field as an independent module that can be coupled to other codes. Thisassumes hydraulic fracture growth to be a local or near well bore phenomenon which is notaffected by far-field reservoir complexities such as reservoir boundaries, faults and barriers.Under this assumption, the two processes: hydraulic fracturing and later production aredecoupled. Thus a local flow problem with appropriate boundary conditions is solved tocompute a local pressure field during fracture propagation.

This forward coupling is computationally inexpensive and adequately captures local flowfield variations effecting fracture growth. Another advantage is that the phase field crackpropagation model generates fracture growth information as a standalone module. Thespatial and temporal scales associated with fracture growth and later production from ahydraulically fractured reservoir are widely different. Therefore, it is reasonable to treat thetwo processes separately. As discussed previously, the phase-field model includes a localized

9

fluid flow description and can therefore generate crack growth information as a stand alone.We then post-process and adapt this crack geometry data for our fractured poroelasticreservoir simulator resulting in a one-way coupling. This approach can be adapted forother legacy reservoir simulators.

Projection of variables/ mesh reconstruction We start with the phase-field approachand solve for p, u, ϕ. At the end of the fracturing process, the reservoir simulator needs thepressure p as initial pressure, ϕ to detect the shape of the fracture and finally the widthw := w(u), which is computed as jump of the normal displacements. The shape of thefracture is determined for all ϕ < thr, where thr denotes a certain threshold, say thr = 0.1(see Figure 4). If ϕ < thr in a cell, it is marked as fracture cell. All unknown quantitiesare computed at cell centers with the associated co-ordinate information to the reservoirsimulator.

Figure 4: Determination of crack shape using threshold of the phase field variable to de-termine fracture cells (marked red). The phase-field module uses locally-refined grids withhanging nodes, which allows to reduce the computational cost significantly.

We post-process and adapt fracture geometry from crack growth model for the fracturedporoelastic reservoir flow model. Fig. 5 shows reconstruction of a coarse, locally distorted,hexahedral mesh which adequately captures the three characteristic length scale variationsof a typical elliptic fracture.

Figure 5: Reconstructing the fracture geometry.

Fig. 6 outlines a work-flow for reconstructing 3D fracture geometry, for reservoir flow

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simulation, from 2D fracture information generated by the fracture growth model. The firstrow shows geometry information for one and three fractures (left and right, respectively)in the YZ plane. A typical fracture growth pattern, in the XZ plane is then used toreconstruct 3D fracture geometries. We use the fact that final fracture geometries after slickwater injection are strongly correlated to reservoir rock property data and can therefore bescaled.

1"

1"

Y

Z

1"

1"

X

Z

Y

X

Z"

1"

1"

1"

Y

X

Z"

1"

1"

1"

1"

1"

Y

Z

Figure 6: Work-flow for reconstructing 3D fracture geometry from 2D information.

Fractured well model As previously described, the crack propagation model alreadycouples fracture flow to reservoir and is therefore complete in itself. That means, startingwith a given setting, we commence with that model and compute a fracture geometry,which might include curvilinear growth, branching and joining. The fracture geometry ispassed to the reservoir simulator, as described in the previous section. The width andthe pressure information from the phase-field model are set as the initial conditions forthe reservoir simulator. Fig. 7 (right) shows fractured well placement (red blocks) in areservoir with natural fractures (shaded orange). The mesh adaption is convenient sinceboth models utilize hexahedral meshes thus avoiding computationally costly interpolationbetween meshes with different mesh elements (tetrahedral, prisms etc.).

The key advantage of our suggested ideas is concerned with the effort in coupling.Rather than iterating in each time step between both frameworks, the phase-field is used asa preprocessor step and as such acts as a own module. This allows us to run different wellplacement scenarios with the reservoir simulator using the same fracture geometry avoidingredundant fracture growth calculations for each scenario. The accuracy of the phase-fieldapproach for modeling fracture propagation increases, as the mesh is refined. The spatialand temporal scales associated with crack propagation are much smaller when compared to

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Figure 7: Integrating fractures generated by phase field model as a fractured well model.

reservoir flow. Therefore, the use of a fine mesh for fracture growth computations followedby reservoir flow calculations on a coarse mesh is computationally efficient. In order toexpedite the calculations for the phase-field fracture growth model, we utilize a dynamicmesh refinement approach with locally refined grids and hanging nodes (see Fig. 4). Forexample, if we run 20 time steps, we perform the first 15 on a coarse mesh and refine thelast 5 time steps to get more accurate fracture tip and associated variable information. Thisprocedure keeps the computational cost very reasonable while increasing accuracy.

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Numerical Tests

We illustrate our methodology by several numerical tests in two and three dimensions.First, we highlight the capabilities of the fractured-well phase-field model and present somecrack propagation scenarios including multi-stage fractures, stress-shadowing effects andcrack growth in heterogeneous porous media with nonplanar fractures. Second, we use oneof these scenarios, extract the fracture and run the reservoir simulator.

The fracture-well phase-field model is computed with the multiphysics template (Wick(2013)) in combination with deal.II’s (Bangerth et al. (2012)) step-31 for the usage of twodifferent degree-of-freedom-handlers to build an iterative solution algorithm as needed forthe fixed-stress splitting. In the following, we provide geometry information and parametersfor the test cases.

Geometry, grid and time step parametersThe computational domain for all 2d tests is Ω = (0, 4)2. Here, two fractures each with

length 1 and midpoints x = 1.5 and x = 2.5 are prescribed. In the second test, the distanceis enlarged and the midpoints are x = 1 and x = 3. In the cases of the three multi-stagefractures, we consider the midpoints x = 1, 2, 3. Here, in the first test all three fractureshave length 1, in the second test the middle fracture has length 0.5 and in the final test1.5. In 3d, in the cube Ω = (0, 10)3, we prescribe a two penny-shape cracks with radiusr = 1.0 in the y = 5.0-plane with mid-points (5.0, 3.0, 5.0) and (5.0, 7.0, 5.0). The crackis approximated as a volume by extending it with the spatial discretization parameter hin up- and downward y-direction, respectively (for details, we refer the reader to Wheeleret al. (2014)). As boundary conditions we set the displacements zero on ∂Ω. We compute50 (2d) and 50 (3d) time steps with time step size ∆t = 0.01 (2d) and ∆t = 0.005 (3d),respectively. We note that the characteristic fracture time scale is,

TF =L2ηF cF

KF

=1× 10−3 × 10−8

10−8= 10−3,

in which we assumed a characteristic fracture length 1 and characteristic fracture perme-ability KF = 10−4.

Model parametersThe augmented Lagrangian penalization parameter is γ = 104 (2d) and γ = 103 (3d).

Several parameters and geometry-related issues depend on the spatial mesh size parameterh. Namely, for the regularization parameters we choose the relations κ = 10−6 × h, ε =2h = 0.088 (2d) and κ = 10−6 × h, ε = 2h = 1.09 (3d).

Flow parameters In all examples, the gravity g is set to zero and the fluid is onlydriven by the point source injection q. We inject fluid at a constant rate into the frac-tures. In 2d and 3d, we use q = 1. Furthermore, the permeability in the reservoir isKR = 10−12. In the the second example, Test 2, we use a randomly varying permeabilitybetween 5 × 10−12 and 10−13. Next, M = 2.5 × 10−8, cF = 10−8, νR = νF = 1.0 × 10−3,ρ0R = ρ0

F = 1. Regarding the Biot coefficient, we perform computations with α = 0 becauseit has been shown in Mikelic et al. (2014a) that α = 0 and α = 1 yield the same crackpatterns if the characteristic time scale of the fracture is taken into account.

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Elasticity and phase-field parametersThe fracture toughness is chosen as Gc = 1.0. The mechanical parameters are µ =

4.2 × 107 and λ = 2.8 × 107. In the second example, we employ randomly varying Lameparameters µ = 4.2× 106 − 9.4× 107 and λ = 2.6× 106 − 9.3× 107.

Comparing fracture propagation in 3D and 2D domains In this example, we firstshow a numerical experiment simulating simultaneous propagation of two penny-shapedfractures in a 3D domain. This is followed by a 2D experiment, in a similar setting, tocompare 2D and 3D results. Fig. 8 shows fracture patterns during growth at T = 0, 15and 25 seconds for the 3D case. Similarly, Fig. 9 shows fracture locations at T = 0, 20and 30 seconds for the 2D case.

Figure 8: Crack pattern for simultaneous propagation of two penny-shaped fractures at T=0, 15 and 25 seconds in 3D domain.

Figure 9: Crack pattern for simultaneous propagation of two fractures at T=0, 20 and 30seconds in a 2D domain.

Since the two cases presented here are symmetrical the temporal variation of pressuresat the centers of the two fractures, for each case, are similar. Fig. 10 shows the timeevolution of pressure at the center of one of the fractures for the 3D (left) and 2D (right)domains. Note that the pressure builds up to threshold value and then starts droppingas the fracture starts growing. The results show resemblance of the fracture growth and

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transient pressure for the 3D and 2D cases.

0

2000

4000

6000

8000

10000

12000

14000

0 10 20 30 40 50

Pre

ssure

Time[5 x 10-3

]

h=0.27

0

5000

10000

15000

20000

25000

0 10 20 30 40 50

Pre

ssure

Time[10-2

]

h=0.044

Figure 10: Transient pressure at the center of the fractures for 3D (left) and 2D (right)cases.

Effect of fracture spacing on fracture growth In this section, we present a numericalexperiment, similar to the 2D case presented earlier, with a larger initial fracture spacingand studying the resulting effect on the fracture pattern. The fracture locations at time T= 0, 20 and 20 seconds are shown in Fig. 11. It can be observed by comparing Figs. 10and 11 that as the spacing is reduced the fracture pattern becomes diverging. This resultdemonstrates that an optimal fracture spacing can be achieved which maximizes reservoirfracture interface area and therefore productivity.

Figure 11: Crack pattern for simultaneous propagation of two fractures, with larger spacing,at T=0, 20 and 30 seconds.

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Effect of discrete fractures on fracture growth In this example, we study the effectof an existing fracture on the propagation of another fracture. This setting is devised toprovide insight into growth patterns for sequential hydraulic fracturing. In Fig. 12 the leftfracture is stationary whereas the right fracture grows due to injection of hydraulic fluids.The stationary fracture (left) is given a higher material stiffness property compared to thereservoir in order to replicate a propped fracture. As it can be seen, the hydraulic fracturedoes not show considerable pattern change due to the presence of an adjacent discretefracture. Although a more detailed study can be conducted to evaluate the combined effectof orientation, we restrict ourselves to the case of parallel fracture for the sake of brevity.Fig. 13 shows the stress fields (Frobenius norm) at times T = 0, 20 and 40 seconds.

Figure 12: Crack pattern for sequential hydraulic fracturing at T=0, 20 and 40 seconds.

Figure 13: Stress field for sequential hydraulic fracturing at T=0, 20 and 40 seconds.

Effect of heterogeneity on fracture growth In this set of tests, we extend the case oftwo simultaneous fracture propagation in a 2D domain to a heterogeneous porous mediaFig. 15 and non-constant reservoir permeabilities Fig. 16. Fig. 15 and Fig. 16 showfracture growth with branching and joining for different times.

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Figure 14: Initial crack pattern (left), randomly distributed Lame coefficients (middle) andnon-constant permeability (right). In the two latter figures, red denotes high values andblue/green low values.

Figure 15: Crack pattern for fracture propagation in a heterogeneous medium at T =20, 30, 50 seconds.

Figure 16: Crack pattern for fracture propagation in a heterogeneous medium and non-constant permeability at T = 5, 10 and 15 seconds.

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Effect of stress shadowing on fracture growth Here, we investigate the effect ofstress-shadowing and initial fracture nucleation lengths on fracture growth for simultane-ous propagation of three fractures. The material properties (Lamee parameters) are kepthomogenous to accentuate observations and are by no means restrictive. Three cases wereconsidered: a) equal fracture nucleation lengths (Fig. 17), b) shorter nucleation length formiddle fracture (Fig. 19) and c) longer nucleation length for middle fracture (Fig. 21).Please note that although boundary conditions play an important role in fracture growththe emphasis here is solely on fracture-fracture interaction.

Figure 17: Example 3, Test 1, crack pattern at T = 0, 20, 30, 50.

Figure 18: Example 3, Test 2, stress distribution at T = 0, 20, 30, 50.

Figs. 18, 20 and, 22 show the stress-fields (Frobenius norm) for the aforementionedthree cases. In, Fig. 17 we observe that the growth of the middle fracture is shunneddue to the stress shadowing from the outer two fractures. Similar behavior is observed forthe case with shorter nucleation length for middle fracture. However, the case with longernucleation length for middle fracture shows contrasting behavior. Here the stress shadowowing to the middle fracture shuns the growth of outer fractures. This numerical test showsthat a careful evaluation of stress shadowing effects is pivotal for planning a hydraulicfracturing job, beginning from perforation to propagation using slick water injection.

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Figure 19: Example 3, Test 2, crack pattern at T = 0, 20, 30, 50.

Figure 20: Example 3, Test 2, stress distribution at T = 0, 20, 30, 50.

Figure 21: Example 3, Test 3, crack pattern at T = 0, 20, 30, 50.

Figure 22: Example 3, Test 3, stress distribution at T = 0, 20, 30, 50.

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TABLE 1—Reservoir properties, Multistage fractures

φ 0.15-0.22 Kx 6= Ky = Kz 0-3800 mDcw 1.E-7 psi−1 co 1.E-4 psi−1

ρw 62.4 lbm/ft3 ρo 56 lbm/ft3νw 1 cP νo 2 cPS0w 0.31 P 0

w 1500 psi

Coupling the phase-field model to a reservoir simulator: a fractured well-boremodel In this section, we present an example to demonstrate the aforementioned approachfor an explicit coupling of fracture growth to a reservoir simulator based upon general hex-ahedral discretization. A synthetic case is generated from Brugge field geometry (see e.g.Peters et al. (2009); Chen et al. (2010)) where the wells are augmented with hydraulic frac-tures. Here the use of fractured wells reduces the number of injection wells while improvingsweep efficiency. The phase field fracture propagation model, followed by production eval-uation of reservoir, allows us to develop an intuitive understanding of recovery predictionsand serves as a decision making tool for design, evaluation and long term field developments.

Figure 23: Coarse fracture mesh after adaptation.

Although not restrictive, for the sake of simplicity, we consider the fracture pattern asshown in Fig. 8. The geometry information from the phase field fracture propagation modelis post-processed and adapted to obtain a coarser mesh while maintaining mesh quality. Thisreduces time-step size restrictions and numerical errors associated with mesh elements. Fig.23 shows the reconstructed, coarse, structured fracture mesh with quadrilateral (hexahedralin 3D) elements. This fracture pattern is integrated with a well-bore model and is used asa fractured well model in our reservoir simulator IPARS.

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Figure 24: Brugge field geometry with fractured injection wells (top) and original Bruggefield case (bottom).

Table 1 provides material and fluid properties required for solving flow and geomechan-ics. The values presented in the table provide typical values used for this simulation run.Fig. 24 (left) shows the fractured Brugge field geometry with 20 bottom-hole pressurespecified production wells at 1000 psi. Here a pressure profile after 2 days is used to aidin visualizing the location of the fractured injection wells. The three red regions show thehydraulically fractured, injection wells with a bottom-hole pressure specification of 2600 psi.The original Brugge field case is shown in Fig. 24 (right) with 30 bottom-hole pressurespecified wells with 10 injectors at 2600 psi and 20 producers at 1000 psi where injectorsare located at a higher elevation compared to the producers. The distorted reservoir ge-

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ometry and fractures are captured using 9×48×139 general hexahedral elements and thendiscretized using a MFMFE scheme (Singh et al. (2014)). Fig. 25 displays permeabilityfields in the X (left) and Y (right) directions. The Z direction permeability is equal to theY direction permeability.

Figure 25: X (left) and Y (right) direction permeability fields.

Fig. 26 shows pressure (left) and saturation (right) profiles at the end of 1000 days.The fractured injection wells are placed at greater depths compared to production wells sothat the gravity assists in oil recovery. A comparison between the pressure and saturationprofiles for the two cases show that a lower number of fractured wells are required forimproved sweep efficiencies compared to conventional wells.

Figure 26: Pressure (left) and saturation (right) profiles after 1000 days for fractured (top)and original (bottom) Brugge field cases.

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Conclusions

In this work, we first presented a fracture propagation model and its numerical solutionscheme for treating slick water hydraulic fracturing based on a phase-field approach. Thisadequately describes crack propagation and associated pressure variation during growth.First, this phase-field model is employed to compute various fracture propagation scenariosin two and three dimensions and crack growth in an anisotropic, heterogeneous, poroelasticmedium. In particular, the pressure profile shows a buildup until a critical energy rateis reached resulting in fracture growth. Second, we successfully coupled this phase-fieldapproach with a reservoir simulator. The integration is based on a computationally efficientone-way coupling which allows the use of the phase-field approach as a pre-processor step.With our proposed approach we are able to simulate hydraulic fracturing and productionstages. An extension to black-oil and compositional models for the reservoir flow descriptioncan also be achieved.

Nomenclature

Ω = reservoir domain

C = fracture domain

E(u, C) = energy functional

σE = stress tensor

e(u) = strain tensor

αB = Biot coefficient

p = fluid pressure

p0 = reference pressure

u = displacement vector

Gc = critical energy release rate

Hd−1(C) = length of fracture, Hausdroff measure

µ, λ = Lame parameters

ν = fluid viscosity

I = identity tensor

ϕ = phase field variable

κ, ε = regularization parameters

w(u) = fracture aperture or width

K = absolutre permeability

ρ = fluid density

g = acceleration due to gravity

qF,R = source/sink for fracture (F) or reservoir (R)

qL = fracture leakage term

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Acknowledgments

This research was funded by ConocoPhillips grant UTA10-000444, DOE grant ER25617,Saudi Aramco grant UTA11-000320 and Statoil grant UTA13-000884. In addition, the firstauthor has been supported by an ICES postdoctoral fellowship and a Humboldt FeodorLynen fellowship. The authors would like to express their sincere thanks for the funding.

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